Prove that $n! > 2^n$ for $n\geq 4$ (solution question)

Question

I'm having a hard time figuring out a part of the solution So I'm trying to prove $n! > 2^n$ for $n \geq 4$

and the solution is attached as a picture

I'm confused as to what happens from the solution line 2 to 3. I don't get the "since k >= 4 and so $k + 1 \geq 2$" reasoning. How does k + 1 become 2 and why does the inequality go from > to $\geq$

score 2 · Accepted Answer · answered Feb 05 '14 at 19:49

2

If $k \ge 4$, then $k+1 \ge 5$ and in particular $k+1 \ge 2$. The switch from $>$ to $\ge$ wasn't really necessary. In fact if $k \ge 4$ then $k+1 > 2$.

answered Feb 05 '14 at 19:49

Umberto P.

52,165

why does it become 2 and not 4? – Jack Feb 05 '14 at 19:54
The idea is to show that $(k+1) \cdot k! > 2\cdot 2^k$. This is true if both $k! > 2^k$ and $k+1 > 2$. – Umberto P. Feb 05 '14 at 20:08

Prove that $n! > 2^n$ for $n\geq 4$ (solution question)

1 Answers1